IV. Choose the proper title to the text.

a. The basic and new concepts             

b. The basic concepts 

c. Modern mathematicians

d. Irrational numbers

V. Give the correct translation of the underlined part.

1. But it is noteworthy, that many more concepts are introduced which are, in essence, creations of human mind.

a) представляют                                        b) были представлены

c) представляли

2. Irrational numbers, negative numbers, and so forth are not wholly abstracted from the physical practice.

a) не отделяются                                        b) не отделяли

c) не были отделены

3. Later concepts are built on earlier notions.

a) строили                                                   b) строятся

c) были построены

4. The basic concepts of the main branches of mathematics are abstractions from experience, implied by their obvious physical counterparts.

a) подразумеваемые                                b) подразумевают

c) подразумевали

5. Irrational numbers, negative numbers and so forth are not wholly abstracted from the physical practice, for the man’s mind must create the notion of entirely new types of numbers to which operations such as addition, multiplication, and the like can be applied.

a) можно будет применить                    b) могли применить

c) могут применяться

Text 12.

Axioms constitute the second major component of any branch of ma­thematics. Up to the XIX century axioms were considered as basic self-evident truths about the concepts involved. We know now that this view ought to be given up. The objective of mathematical activity consists of the theorems deduced from a set of axioms. The amount of information that can be deduced from some sets of axioms is almost incredible. The axioms of number give rise to the results of algebra, properties of func­tions, the theorems of the calculus, the solutions of various types of differential equations. Mathematical theorems mist be deductively establish­ed and proved. Much of the scientific knowledge is produced by deductive reasoning; new theorems are proved constantly, even in such old sub­jects as algebra and geometry and the current developments are as im­portant as the older results.

Growth of mathematics is possible in another way. Mathemati­cians are sure now that sets of axioms which have no bearing on the phy­sical world should be explored. Accordingly, mathematicians nowadays investigate algebras and geometries with no immediate applications. There is, however, some disagreement among mathematicians as to the way they answer the question: “Do the concepts, axioms, and theorems exist in some objective world and are merely detected by man or are they entirely human creations?” In ancient times the axioms and theorems were regarded as necessary truths about the universe already incorporated in the design of the world. Hence each new theorem was a discovery, a disclosure of what already existed. The contrary view holds that ma­thematics, its concepts, and theorems are created by man. Man distin­guishes objects in the physical world and invents numbers and number names to represent one aspect of experience. Axioms are man’s generali­zations of certain fundamental facts and theorems may very logically follow from the axioms. Mathematics, according to this view-point, is a human creation in every respect. Some mathematicians claim that pure mathematics is the most original creation of the human mind.

http://abc.vvsu.ru/Books/engl_matem